The efficiency of a random and fast switch in complex dynamical systems

In this paper, we show that a fast switch is able to lead a complex dynamical system to being asymptotically stable, although this system is completely unstable in every switch duration and even the associated connection matrices are randomly selected. Importantly, we define some new exponents by which we can figure out the essential patterns that guarantee the stability of fast switching systems, and besides, their calculations need little computational cost. More interestingly, we show the efficiency of some random switches in inducing stability through a comparison of the systems with different switch connection matrices and switch durations, and we give a design method for obtaining higher efficient random switch rules. We also investigate the generalization of the obtained results to a more realistic case where the switch obeys some renewal process.

stability in fast switching systems. Indeed, in each switch duration, the random dynamical system can be completely unstable and the connection matrices can take values along some continuously valued probability density function (PDF). More interestingly, through a comparison, the systems with randomly switching connection matrices are found to be much easier to stabilize or synchronize, where the form of the corresponding PDF plays a crucial role. We give a design method for obtaining higher efficient random switch rules. We also discuss the generalization of the obtained results to the case when the switch obeys some kind of renewal process [25,26].

Examples showing the efficiency of a random switch
To begin, we consider the Lorenz systeṁ x = 10(y − x), y = x(28 − z) − y, z = x y − 8z/3 with a simple and linear feedback control: F = S q (x, y, z) , where S q is a 3 × 3 gain matrix selected from a matrix set M e = {S e 1 , S e 2 , S e 3 } and q = 1, 2, 3. Without the control, the Lorenz system exhibits chaotic dynamics, as shown, respectively, in the time periods I 2k−1 (k = 1, 2, . . . , 5) in figure 1. With a static control, i.e. the gain matrix S q is fixed as either one of the three matrices in M e all the time, the controlled Lorenz system is stabilized to some equilibria, as shown, respectively, in the periods I 2 , I 4 and I 6 in figure 1. However, for the origin (x = y = z = 0), the static control is not stable yet, since its instability can be verified simply by a positive eigenvalue of the matrix DL(0) + S e q , the Jacobian matrix of the controlled Lorenz system around the origin. Also in figure 1, with a deterministic-switching control, i.e. the gain matrix is selected periodically among S e 1 , S e 2 and S e 3 with a switch duration 0.001, the controlled Lorenz system is stabilized to the equilibrium in the period I 8 . Nevertheless, this stabilized equilibrium is not the steady state of the uncontrolled Lorenz system, and the origin is still unstable. Strikingly, as shown in the period I 10 in figure 1, the unstable origin can be indeed stabilized when the random-switching control is adopted, i.e. the gain matrix is selected stochastically among S e 1,2,3 with respective probabilities p 1,2,3 and a switch duration 0.001. Secondly, we consider the synchronization among three coupled Lorenz systems: where each state variable x i = (x i , y i , z i ) and L is the vector field of the Lorenz system as shown above. The coupling matrix S q = {S i j q } is a 3 × 3 Laplacian matrix selected from a matrix set M s = {S s 1 , S s 2 }. As shown in figure 2, in the first two time durations J 1,2 , the synchronization cannot be realized when the connection matrix S q is fixed as one of the matrices in M s . Either matrix in M s therefore belongs to unstable coupling. Furthermore, the synchronization cannot be surely achieved in the time duration J 3 if S q is selected periodically between S s 1 and S s 2 with a switch duration 0.01. Interestingly, the chaotic synchronization can be realized in the last duration J 4 if S q is selected stochastically between S e 1,2 with respective probabilities p 1,2 as shown in the caption of figure 2. Here, the random switch shows the special capability of realizing chaotic synchronization among coupled nonlinear systems.

Mathematical configuration of the model
The above simulation results show the capability and efficiency of a random switch among unstable controls in the stabilization or synchronization of complex dynamical systems to an unstable steady state or chaotic state of the original system. To fully illustrate the phenomenon observed above, we consider an m-dimensional complex dynamical system which is described byẋ with an initial condition x(0) = x 0 ∈ R m . Here σ : R + → N is an index function of the switching rule, which can be either random or deterministic. Moreover, the connection matrix S σ (t) describes the switching interactions among the m-components of the system, and it takes a matrix value S i k ∈ R m×m when the index σ (t) ≡ i k ∈ N and t belongs to the switch duration During each switch duration, system (1) becomes simply linear and the interaction matrix can take a matrix value either randomly or deterministically.
For example, the linearization around the origin of the controlled Lorenz system we considered above can be regarded as a particular case of system (1), when the connection matrix S σ (t) in system (1) is set as the Jacobian matrix of the controlled Lorenz system around origin, namely S σ (t) = DL(0) + S e q . Here, the index function σ (t) = q for t ∈ I k and t k = 0.001k, and S e q is the gain matrix as defined above. More concretely, when the static control is used for the Lorenz system, σ (t) = q is fixed as one of the three numbers {1, 2, 3} for all t ∈ [0, +∞). When the deterministic-switching control is considered, the periodic index function is taken as σ (t) = q = 1 + (k mod 3) for t ∈ I k and all k ∈ N. When the random-switching control is adopted, σ (t) = q in each I k takes a value from {1, 2, 3} with respective probabilities p 1,2,3 . Moreover, like the coupled Lorenz systems we investigated above and by adopting the variational technique along some steady synchronization manifold [1,4,[21][22][23][24], most complex dynamical network systems of physical relevance which are described by the linear or nonlinear dynamical systemṡ could also be formulated into some kind of switching linear systems as system (1). For these systems, the eigenvalues of the coupling matrix S i j σ (t) contribute to the finally formulated connection matrix S σ (t) as defined in system (1). In addition, May's model [7] could be even regarded as a specific case of system (1) since for his model every i k in system (1) is identical, but each element of the identical S i k = S is selected randomly.
To investigate the asymptotical stability of the switching system (1), we write the solution of this system as where I is an identical matrix and F i k needs to be estimated later. Also, in what follows, we need a matrix decomposition: where D i k stands for the diagonals of S i k and the other elements constitute A i k whose diagonals are all 0. This decomposition plays a crucial role in establishing an exponent as well as criteria for guaranteeing the stability in sufficiently fast switching systems (1). With this decomposition, each matrix e S i k T k can be further written as In what follows, we are to establish feasible conditions that guarantee the asymptotical stability of system (1), that is, x(t) → 0 as t → +∞ in the sense of probability one for a random complex dynamical system, or x(t) → 0 as t → +∞ for a deterministic system.

The case of random complex systems
For a random complex dynamical system, we consider that the elements of S i k are randomly and independently selected in each switch duration I k . More technically, for each fixed i k , S i k is supposed to be a random and independent matrix S(ω) = D(ω) + A(ω), and thus its elements S i j (ω) (or D i j (ω) and A i j (ω)) become random variables on the probability space ( 0 , F 0 , P). In particular, ω ∈ 0 and each S i j (·) maps 0 into an interval J = [−M, M] ⊂ R. For example, the blinking complex networks investigated in [19,20] are of specific switching systems with each S i j (ω) taking two discrete values with probabilities p i j and 1 − p i j . Indeed, each S i j (ω) is allowed to take multiple or continuously distributed values, obeying some PDF ρ(S i j ) whose compact support is J . In fact, this kind of interaction among the components is even ubiquitous in the real world. Now, we are in a position to establish the conditions that ensure asymptotical stability of the solution of system (1) in the physical sense (i.e. in the sense of probability one). First, as mentioned above, we make an estimation (E): where · denotes the Euclidean norm, and the last inequality holds if the upper bound of the switch duration T is sufficiently small, such that T < T * = (m M) −1 . Secondly, for the random system (1), {x k } k∈N , set above, can be regarded as a stochastic process on ( 0 , F k , P), where F k , the natural filtration of x k , satisfies F k−1 ⊂ F k for all k. Thus, we estimate the conditional expectation as follows: |A ji |] since in every duration I k the elements of the connection matrix obey the same probability distributions as assumed above. Moreover, the independence property of the random matrix selected in each duration I k and T < T * are used for obtaining the formula after the third 7 equality, and the estimation (E) ensures the formula after the second inequality. Therefore, the nonnegative stochastic process { x k 1 } k∈N becomes a super-martingale provided that α ∈ (0, 1), that is, conditions (RS): χ r < 0 and are satisfied. Indeed, with these conditions and according to the theory of martingale [25,26], x k 1 is convergent to a nonnegative random variable, denoted by ζ , in the sense of probability one. Note that from the above estimation. Therefore, through recursive arguments and from Fatou's lemma [25,26], it follows that the random variable ζ = 0 in the sense of probability one, which therefore yields a conclusion that the asymptotical stability of the solution x(t) produced by the random system (1) is ensured in the physical sense if conditions (RS) are satisfied.
Here, we further provide some illustrations of conditions (RS). Note that the exponent χ r represents the expectation of some combination of the connection matrix elements. The condition χ r < 0 thus means, from the viewpoint of probability, that the diagonal elements need to be negatively dominant in columns; in other words, the self-interacting intensity of each component in a complex system should be negatively stronger on average than the summation of its incoming interacting intensities. Generally, the larger the size of the system, the stronger the self-interacting intensity that is required to satisfy the condition. As is well known, for a nonrandom matrix, diagonally negative dominance implies that its eigenvalues are all located in the left-half complex plane. Nevertheless, the condition χ r < 0 we established above does not lead surely to this kind of eigenvalues distribution for the matrix randomly selected in every duration I k . Indeed, it allows that the random matrix be unstable for almost every i k . For example, consider a complex system with a 2 × 2 random connection matrix S(ω) in which the diagonals {S 11 , S 22 } = {ξ 11 , ξ 22 } + η and S i j = ξ i j for i, j = 1, 2 and i = j. Here, ξ i j (ω) are mutually independent and continuous random variables, which obey the uniform distribution on [0, 1]. The function η(ω) is a discrete random vector taking values {−θ, 0.1} and {0.1, −θ }(θ > 0), respectively, with probabilities p and 1 − p. For this example, the abovedesigned exponent Hence, for each ω, the matrix S(ω) is unstable with at least one positive eigenvalue, which is numerically verified in figure 3(a) for θ = 3 and p = 0.5. Therefore, in such a case, the random dynamical system (1) is unstable in every switch duration I k . However, according to the conclusion obtained above, we can still realize the asymptotical stability in the physical sense for such a system provided that the other condition T <T is fulfilled ( figure 3(b)). Interestingly, this shows that a sufficiently fast switch can induce stability in some kind of not only unstable but also random complex dynamical systems.
Furthermore, the analytically obtained upper boundT is strongly dependent on the system size m as well as on the maximal range M of all the random variables. The larger the size of the system, the faster the switching speed required for asymptotical stability. Figure 3(c) further shows this relation numerically, which is qualitatively consistent with the analytical bounds. Also, it is mentioned that the proposed exponent χ r , as well as the upper boundT , needs little computational cost if the probability distributions for the elements of the connection matrix are specified. In particular, we can compute the exponent directly through as long as ρ i j , the a priori or a posteriori PDFs on the interacting intensities among the components in real systems, are given. Hence, determination of the stability of a fast switching system no longer needs a high-cost computation of the matrix eigenvalues or the Lyapunov exponents over an uncertain period of time [12][13][14].

The case of deterministic complex systems
For a deterministic system, the switching rule of the connection matrix is fixed in advance. Accurately, S σ (t) takes values along the sequence S i k in every switch duration I k for k ∈ N, where both sequences {I k } and {i k } are given deterministically, and S i k takes values from an allowable matrix set {S n } n∈I that can contain a finite or infinite number of elements. For example, consider an allowable matrix set that contains only two unstable matrices: S 1 = [1 1; 1 −4] and S 2 = [−4 1; 1 1], so that I = {1, 2}. Set I k = [k − 1, k) and i k = 1 + (k mod 2) for k ∈ N. Then, the constructed switching rule makes system (1) switch periodically between the two unstable subsystems. 9 Now, a question arises: 'Can we still stabilize a system of this kind when the speed of the deterministically-fixed switch is adjusted to be sufficiently fast?'. Although the answer to this question is positive, which could be found in [12][13][14], we, through a particular argument below, establish some more direct and relaxable conditions to answer the question. Only with the conditions established below we are able to make a comparison and identify which kind of switch is more efficient at stabilization. Hence, we consider the other system as follows: where the smaller the value ofT (<1), the faster the switching speed. System (2) therefore could be regarded as an accelerated model of system (1). In the following discussion, suppose the original duration T k ∈ [η, 1] for simplicity; otherwise, we can insert some time instants without changing the real switching rules. Denote the switch durationT T k of system (2) by T d k . Then T d k ∈ ηT ,T . For system (2), we still adopt the matrix decomposition utilized for system (1) and analogously implement the argument performed in the estimation (E). Then, we obtain max 1 i, j m |F i j i k | e δ − 1 − δ ifT < T δ = δ(max n∈I S n 2 ) −1 and δ 1, so that the decomposed matrices satisfy Here and throughout, X M = max n∈I,1 i, j m |X i j n | and X n denotes a matrix. We further estimate the solution of system (2) around t = t (k+1)N with some integer N as follows: where we need two conditions: namely condition (DS1): for any integer k k 0 0 and some constant c > 1 with cζ 1 < 1, and condition (DS2): withT < T δ . The notation h X h above represents an insertion of the matrix X h into the position h of the matrix multiplication in front of this notation. Using the above estimation technique Next, we interpret the two conditions (DS1) and (DS2). On the one hand, condition (DS1) can be transformed as for all integer k k 0 . Here, the negativity of the exponent χ d implies that the time average of the matrix diagonals along the allowable connection matrix set is required to be sufficiently negative. This condition is somewhat analogous to χ r < 0 for the random complex system, where, however, not A M but the average absolute value of the off-diagonal entries is taken into account. Moreover, the smallest switch duration η, named the dwell time of system (1), also restricts the value of χ d . Thus, the condition for deterministic systems is somewhat stronger than that for random systems. Note also that the negativity of the maximal time average does not necessarily imply the stability of each allowable connection matrix. Still consider the example provided earlier in this section. Correspondingly, η = A M = 1 and m = N = 2. Therefore, the exponent χ d = −0.5 < 0, although both connection matrices are unstable.
On the other hand, condition (DS2) seems to be complicated; however, it actually implies a sufficiently small value ofT . To intuitively see this, we depict in figure 4(a) the interval of T with the parameters of the above-imported example. Although this interval is narrow, with χ d < 0, it implies that we are able to stabilize the system if the speed of the deterministic switch is sufficiently fast ( figure 4(b)). Therefore, the question posed above is answered completely. probabilities for the matrices are given by p, q and 1 − p − q, the stability region for these probabilities is depicted analytically through conditions (RS), shown by the triangle region in figure 5(a). However, as shown in figure 5(b), the stability cannot be realized in the system that switches periodically among the three matrices S 1,2,3 . Thus, stability induced by fast switching is more easily realized in random complex systems. In fact, as the switch speed increases, some random switches are capable of inducing the system trajectory to more frequently visit the stable manifold generated by a proper combination of the connection matrices, which therefore leads to the negatively dominant diagonals in the probability sense. Nevertheless, the deterministic switch rarely possesses such an advantage when the switching order of the connection matrices is specified in advance. Also in figure 5(a), the stability region through the numerical experiment for all 500 times is depicted. The numerically obtained region is broader than the analytical region since conditions (RS) are only sufficient. In addition, not every but a proper designed PDF is suitable for stabilizing the random complex system. As shown in figures 5(a) and (b), even the simple uniform probabilities cannot ensure the stability. Finally, the above analyses also illustrate why the particularly designed random-switching control is capable of stabilizing the Lorenz system to the unstable origin in the previous section.

The switch obeying a renewal process
In the real world, the switch often obeys some renewal process, unlike the bounded switch duration used in the above discussion. For example, those intervals between potential spikes emitted by neurons usually satisfy either the Poisson distribution or some other distributions [5,27]. In fact, all the results can be generalized analytically to such a case if some additional conditions on the PDF of the process are taken into account. The detailed and analytic generalization will be included in our future work. Here, to illustrate it numerically, we consider the example used in figures 3(a) and (b). Instead, T k of the switch is supposed to be a renewal process obeying the PDF specified in figure 5(c). As shown in figure 5(d), the asymptotical stability can be realized in the physical sense if there exists a number T c ∼ M −1 such that +∞ T c e m M T ρ(T )dT is sufficiently small. This means that to obtain the stability, the switch needs to be sufficiently fast most of the time and is allowed to be slow occasionally.

Concluding remarks
We have defined some exponents by which we are able to analytically identify the essential patterns that guarantee the stability or synchronization of complex dynamical systems with either a deterministically or a randomly switching configuration. The analysis of these exponents has allowed us to describe the capacity and efficiency of the random switch for stabilizing or synchronizing a complex dynamical system which is indeed unstable in every switch duration. Also we have emphasized the possibility of generalizing the obtained results to the more realistic case when the switch obeys some kind of renewal process. We believe that all the results obtained in this paper are of practical use for the analysis and modulation of the dynamical behaviors of some real complex systems involving random fluctuations.